A company's daily profit, in dollars, is modeled by P(x) = -3x² + 150x - 800, where x is the number of units sold. How many units should…

The profit function $P(x) = -3x^2 + 150x - 800$ is a downward-opening parabola, so its maximum is at the vertex. The vertex has $x = -\frac{b}{2a} = -\frac{150}{2(-3)} = 25$ units. The constant $-800$ shifts the profit up or down but does not change where the maximum occurs. The maximum profit is $P(25) = -1875 + 3750 - 800 = 1075$ dollars. A common wrong answer is 1075, which is the maximum profit rather than the number of units. Another is 50, from using $-\frac{b}{a}$, forgetting to double the denominator. Answer the quantity asked.